Complex Digital Filters Research Articles

Principles of complex wave digital filters

AbstractThe theory of real wave digital filters (WDFs) is extended to complex WDFs. A fundamental requirement turns out to be that of one‐realness, a complex impedance in the reference filter being called one‐real if it is real for Ψ = 1, where Ψ is the equivalent complex frequency. The only new components then required are complex multipliers with unimodular coefficients, but the adaptors are just as in the real case. The stability properties known for real WDFs can, in this way, also be extended.

Complex Convolutions via Fermat Number Transforms

An approach is described for computing complex convolutions modulo a Fermat number. It is shown that this technique is particularly efficient when the complex convolution is computed by means of Fermat Number Transforms and leads to improved implementation of complex digital filters.

Design of MTI detection filters with nonuniform interpulse periods

This paper presents an analysis of the transfer function of digital MTI detection filters of the type <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(1-z^{-1})^{n}</tex> followed by a squarer and a detector where the designer has the freedom to vary the interpulse periods. This degree of freedom produces some interesting and useful results, as these simple filters can then be made to exhibit transfer-function characteristics similar to more complex recursive digital filters, yet retaining the finite transient characteristics of the simple nonrecursive filter. Furthermore, there is a great reduction in hardware as the simple filters require no multipliers and fewer storage registers. The key to the analysis and design of such a pulse-staggered filter is the recognition that the magnitude squared transfer function represents a frequency-interference pattern. By properly arranging the interpulse periods the nulls of the interference pattern will coincide with the desired nulls of the required filter-response characteristic. Assuming an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N + 1</tex> pulse burst ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> odd) to the digital filter, it is shown that the normalized transfer function of the filter <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1-z^{-1}</tex> is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">|H_{N}(\omega)|^{2}=1- \frac{1}{N}\Sum_{i=1}^N \cos \omega T_i</tex> . <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T_{i}</tex> is ith interpulse period. If the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T_{i}</tex> are arranged such that they are uniformly spaced about a mean TA with spacing T, then this transfer function becomes the interference pattern <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1- \left[\frac{\sin \frac{N\omega T}{2}}{\sin \frac{\omega T}{2}}\right]</tex> The value of T is adjusted for the desired true nulls of the frequency characteristic, N is adjusted for the ripple in the passband, and TA is adjusted for mimum number of extrema and cut-off characteristics in the stopband. The transfer characteristic of the digital filter (l-z-')' exhibits two interference patterns, one corresponding to E cos coTe and one corresponding to E cos co(Ti + Ti+,). In this case the order of the Ti is crucial, as large nulls in the passband are to be avoided. Consequently, the Ti are again arranged uniformly spaced about TA with spacing T, but they are rearranged such that the sequence of the sum of two successive interpulse periods have a difference equal to T. When this is accomplished, the main lobes of the modulating envelopes of the interference patterns will coincide only at integer multiples of I/ T, resulting in a passband. Both a mathematical analysis and engineering design rules are presented in this paper.